In grades 5-8, the
mathematics curriculum should include the study of number systems
and number theory so that students can--
- understand and
appreciate the need for numbers beyond the whole numbers;
- develop and use
order relations for whole numbers, fractions, decimals, integers,
and rational numbers;
- extend their understanding
of whole number operations to fractions, decimals, integers, and
- understand how
the basic arithmetic operations are related to one another;
- develop and apply
number theory concepts (e.g., primes, factors, and multiples)
in real-world and mathematical problem situations.
The central theme of this
standard is the underlying structure of mathematics, which bonds
its many individual facets into a useful, interesting, and logical
whole. Instruction in grades 5-8 typically devotes a great deal
of time to helping students master a myriad of details but pays
scant attention to how these individual facets fit together. It
is the intent of this standard that students should come to understand
and appreciate mathematics as a coherent body of knowledge rather
than a vast, perhaps bewildering, collection of isolated facts and
rules. Understanding this structure promotes students' efficiency
in investigating the arithmetic of fractions, decimals, integers,
and rationals through the unity of common ideas. It also offers
insights into how the whole number system is extended to the rational
number system and beyond. It improves problem-solving capability
by providing a better perspective of arithmetic operations.
Instruction that facilitates
students' understanding of the underlying structure of arithmetic
should employ informal explorations and emphasize the reasons why
various kinds of numbers occur, commonalities among various arithmetic
processes, and relationships between number systems. Steen
(1986, p. 6), past president of the Mathematical Association
of America, articulated the spirit of this standard when he noted
that "above all else, it [the mathematics curriculum] must
not give the impression that mathematical and quantitative ideas
are the product of authority or wizardry."
Number theory offers many
rich opportunities for explorations that are interesting, enjoyable,
and useful. These explorations have payoffs in problem solving,
in understanding and developing other mathematical concepts, in
illustrating the beauty of mathematics, and in understanding the
human aspects of the historical development of number.
As students reach grade
5, they begin to recognize--in both arithmetic and geometric settings--the
need for numbers beyond whole numbers.
Arithmetically, the fraction
2/3 becomes necessary as the only solution to the whole number problem
2 ÷ 3, such as in the real-world situation in which two pizzas
are divided among three people. The integer - 1 becomes necessary
so that the problem 2 - 3 has a solution, such as when a player
loses three points in a game when he or she has only two points.
The need to measure more precisely than to the nearest inch gives
rise to numbers like 3 5/8 inches. Such numbers as the 2
and are needed to describe the
length of the diagonal of a square or the circumference of a circle.
Encountering irrationals as nonexamples helps students appreciate
As students expand their
mathematical horizons to include fractions, decimals, integers,
and rational numbers, as well as the basic operations for each,
they need to understand both the common ideas underlying these number
systems and the differences among them. For example, to compare
2/3 and 3/4, students can use concrete materials to represent them
as 8/12 and 9/12, respectively, and then to conclude that 8/12 is
less than 9/12, since 8 is less than 9. Thus, they learn that comparing
fractions is like comparing whole numbers once common denominators
have been identified. Yet in comparing -2 and -5 on a number line
or a thermometer, students see that -2 is greater than -5, so they
learn that comparing negative integers is different from comparing
Students should extend
their knowledge of whole number operations other systems; for example,
they need to see how 1.4 + 6.7 is related to 14 + 67 and how dividing
- 12 by - 3 is related to dividing 12 by 3. Concrete materials and
representational models, such as area models, should be used to
provide a firm basis for understanding such ideas. The transition
from whole numbers to fractions and decimals can be difficult for
students. Although they may multiply the numerators and then the
denominators, for example, they often do not understand why a similar
procedure does not work in adding fractions. Concrete or representational
models can help students clarify these anomalies.
Students should understand
how analogies among structures can give a clearer picture of mathematics.
For example, in contrasting the missing-addend interpretation of
subtraction with the missing-factor interpretation of division,
students learn that
12 - 3 = n means
the same thing as 3 + n = 12
12 ÷ 3 = n
means the same thing as 3 x n = 12.
This suggests that subtraction
can be considered in terms of addition and that division can be
thought of in terms of multiplication, illustrating the analogy
that division is to multiplication as subtraction is to addition.
Relationships among operations
are developed over many grades. In K-4, multiplication can be viewed
as repeated addition and division can be considered repeated subtraction.
However, because the multiplication of fractions and decimals is
not repeated addition and division is not always repeated subtraction,
students should be exposed to other interpretations of multiplication
and division as well. For example, as shown in figure
5.1, 1.2 x 1.3 can be represented by an area model. In the later
grades, students learn that the subtraction of integers is the same
as adding the opposite and the division of fractions is the same
as multiplying by the reciprocal.
Another goal of this standard
is to offer meaningful answers to such questions as, Why can't we
divide by zero? Is there a smallest number? A largest number?
Challenging but accessible
problems from number theory can be easily formulated and explored
by students. For example, building rectangular arrays with a set
of tiles can stimulate questions about divisibility and prime, composite,
square, even, and odd numbers. (See fig. 6.1.)
This activity and others
can be extended to investigate other interesting topics, such as
abundant, deficient, or perfect numbers; triangular and square numbers;
cubes; palindromes; factorials; and Fibonacci numbers. The development
of various procedures for finding the greatest common factor of
two numbers can foreshadow important topics in the 9-12 curriculum,
as students compare the advantages, disadvantages, and efficiency
of various algorithms. String art and explorations with star polygons
can relate number theory to geometry.
Another example from number
theory involves making connections between the prime structure of
a number and the number of its factors:
Find five examples of
numbers that have exactly three factors. Repeat for four factors,
then five factors. What can you say about the numbers in each of
Students might give 4,
9, 25, 49, and 121 as examples of numbers with exactly three factors.
Each of these numbers is the square of a prime.
Without an understanding
of number systems and number theory, mathematics is a mysterious
collection of facts. With such an understanding, mathematics is
seen as a beautiful, cohesive whole.